C-Spline Basis for Polynomial Splines

Generates the convex regression spline (called C-spline) basis matrix by integrating I-spline basis for a polynomial spline or the corresponding derivatives.

Usage

cSpline(
  x,
  df = NULL,
  knots = NULL,
  degree = 3L,
  intercept = TRUE,
  Boundary.knots = NULL,
  derivs = 0L,
  scale = TRUE,
  ...
)

Arguments

x: The predictor variable. Missing values are allowed and will be returned as they are.
df: Degree of freedom that equals to the column number of the returned matrix. One can specify df rather than knots, then the function chooses df - degree - as.integer(intercept) internal knots at suitable quantiles of x ignoring missing values and those x outside of the boundary. If internal knots are specified via knots, the specified df will be ignored.
knots: The internal breakpoints that define the splines. The default is NULL, which results in a basis for ordinary polynomial regression. Typical values are the mean or median for one knot, quantiles for more knots.
degree: The degree of C-spline defined to be the degree of the associated M-spline instead of actual polynomial degree. For example, C-spline basis of degree 2 is defined as the scaled double integral of associated M-spline basis of degree 2.
intercept: If TRUE by default, all of the spline basis functions are returned. Notice that when using C-Spline for shape-restricted regression, intercept = TRUE should be set even when an intercept term is considered additional to the spline basis in the model.
Boundary.knots: Boundary points at which to anchor the splines. By default, they are the range of x excluding NA. If both knots and Boundary.knots are supplied, the basis parameters do not depend on x. Data can extend beyond Boundary.knots.
derivs: A nonnegative integer specifying the order of derivatives of C-splines. The default value is 0L for C-spline basis functions.
scale: A logical value indicating if scaling C-splines is required. If TRUE by default, each C-spline basis is scaled to have unit height at right boundary knot. The corresponding I-spline and M-spline produced by deriv methods will be scaled to the same extent.
...: Optional arguments that are not used.

Value

A numeric matrix of length(x) rows and df columns if

df is specified or length(knots) + degree + as.integer(intercept) columns if knots are specified instead. Attributes that correspond to the arguments specified are returned mainly for other functions in this package.

Details

It is an implementation of the closed-form C-spline basis derived from the recursion formula of I-splines and M-splines.

References

Meyer, M. C. (2008). Inference using shape-restricted regression splines. The Annals of Applied Statistics, 2(3), 1013--1033.

Examples

library(splines2)

x <- seq.int(0, 1, 0.01)
knots <- c(0.3, 0.5, 0.6)

### when 'scale = TRUE' (by default)
csMat <- cSpline(x, knots = knots, degree = 2)

op <- par(mar = c(2.5, 2.5, 0.2, 0.1), mgp = c(1.5, 0.5, 0))
matplot(x, csMat, type = "l", ylab = "C-spline basis")
abline(v = knots, lty = 2, col = "gray")

isMat <- deriv(csMat)
msMat <- deriv(csMat, derivs = 2)
matplot(x, isMat, type = "l", ylab = "scaled I-spline basis")

matplot(x, msMat, type = "l", ylab = "scaled M-spline basis")


## reset to previous plotting settings
par(op)

### when 'scale = FALSE'
csMat <- cSpline(x, knots = knots, degree = 2, scale = FALSE)

## the corresponding I-splines and M-splines (with same arguments)
isMat <- iSpline(x, knots = knots, degree = 2)
msMat <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)

## or using deriv methods (more efficient)
isMat1 <- deriv(csMat)
msMat1 <- deriv(csMat, derivs = 2)

## equivalent
stopifnot(all.equal(isMat, isMat1, check.attributes = FALSE))
stopifnot(all.equal(msMat, msMat1, check.attributes = FALSE))